2016/07/16 by Franklin D. Tall, Tall, Franklin D.
Computer Science · Mathematics · #03E15 #03E35 #03E60 #03E75 #54A35 #54D45 #54H05 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #General Topology (math.GN) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1607.04781
openalex publication_date 2016/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Menger's conjecture that Menger spaces are /sigma-compact is false; it is true for analytic subspaces of Polish spaces and undecidable for more complex definable subspaces of Polish spaces. For non-metrizable spaces, analytic Menger spaces are /sigma-compact, but Menger continuous images of co-analytic spaces need not be. The general co-analytic case is still open, but many special cases are undecidable, in particular, Menger topological groups. We also prove that if there is a Michael space, then productively Lindelof Cech-complete spaces are /sigma-compact. We also give numerous characterizations of proper K-Lusin spaces. Our methods include the Axiom of Co-analytic Determinacy, non-metrizable descriptive set theory, and Arhangel'skii's work on generalized metric spaces.